cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A360100 a(n) = Sum_{k=0..n} binomial(n+2*k-1,n-k) * Catalan(k).

Original entry on oeis.org

1, 1, 5, 23, 111, 562, 2952, 15948, 88076, 495077, 2823293, 16295020, 95007654, 558765743, 3310999269, 19748462718, 118471172054, 714355994997, 4327148812557, 26319195869861, 160677354596769, 984236344800234, 6047526697800992, 37262944840704171
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums are A360102.
Cf. A162481, A258973.
Cf. A002212, A006319, A162475, A360101.
Cf. A000108.

Programs

  • Maple
    A360100 := proc(n)
    add(binomial(n+2*k-1,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360100(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • Mathematica
    m = 24;
A[_] = 0;
Do[A[x_] = 1 + x A[x]^2/(1 - x)^3 + O[x]^m // Normal, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Aug 16 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n+2*k-1, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/(1+sqrt(1-4*x/(1-x)^3)))

Formula

G.f. A(x) satisfies A(x) = 1 + x * A(x)^2 / (1-x)^3.
G.f.: c(x/(1-x)^3), where c(x) is the g.f. of A000108.
a(n) ~ sqrt(-2 + (35 - 3*sqrt(129))^(1/3) + (35 + 3*sqrt(129))^(1/3)) * (((7 + (262 - 6*sqrt(129))^(1/3) + (2*(131 + 3*sqrt(129)))^(1/3))/3)^n / (sqrt(2*Pi) * n^(3/2))). - Vaclav Kotesovec, Feb 18 2023
D-finite with recurrence (n+1)*a(n) +(-8*n+5)*a(n-1) +(10*n-27)*a(n-2) +(-4*n+17)*a(n-3) +(n-6)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A364623 G.f. satisfies A(x) = 1/(1-x)^3 + x*A(x)^3.

Original entry on oeis.org

1, 4, 18, 112, 847, 7086, 62974, 583002, 5560323, 54249583, 538873135, 5431177821, 55402340842, 570899082760, 5933922697380, 62138800690564, 654949976467593, 6942859160218698, 73972792893687427, 791722414873487767, 8508265804914763731
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Links

Crossrefs

Partial sums of A364629.
Cf. A162481, A364624.
Cf. A001764, A199475, A364620.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+5*k+2, 6*k+2)*binomial(3*k, k)/(2*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+5*k+2,6*k+2) * binomial(3*k,k) / (2*k+1).

A360057 a(n) = Sum_{k=0..n} binomial(n+4*k+4,n-k) * Catalan(k).

Original entry on oeis.org

1, 6, 27, 125, 644, 3643, 21974, 138395, 898695, 5970927, 40386209, 277127148, 1924349756, 13496536510, 95467320600, 680260392219, 4878382821267, 35182209381590, 255000022472565, 1856501085686340, 13570366067586294, 99554601986349471, 732756800760507312
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A086616, A162481, A358518.
Cf. A000108, A360047.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+4*k+4, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^5*(1+sqrt(1-4*x/(1-x)^5))))

Formula

a(n) = binomial(n+4,4) + Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^5 + x * A(x)^2.
G.f.: 2 / ( (1-x)^5 * (1 + sqrt( 1 - 4*x/(1-x)^5 )) ).
D-finite with recurrence (n+1)*a(n) +2*(-5*n+1)*a(n-1) +(19*n-11)*a(n-2) +20*(-n+2)*a(n-3) +15*(n-3)*a(n-4) +6*(-n+4)*a(n-5) +(n-5)*a(n-6)=0. - R. J. Mathar, Jan 25 2023

A360102 a(n) = Sum_{k=0..n} binomial(n+2*k,n-k) * Catalan(k).

Original entry on oeis.org

1, 2, 7, 30, 141, 703, 3655, 19603, 107679, 602756, 3426049, 19721069, 114728723, 673494466, 3984493735, 23732956453, 142204128507, 856560123504, 5183708936061, 31502904805922, 192180259402691, 1176416604202925, 7223943302003917, 44486888142708088
Offset: 0

Views

Author

Seiichi Manyama, Jan 25 2023

Keywords

Crossrefs

Partial sums of A360100.
Partial sums are A258973.
Cf. A000108, A162481.
Cf. A006318, A007317, A162476, A360103.

Programs

  • Maple
    A360102 := proc(n)
    add(binomial(n+2*k,n-k)*A000108(k),k=0..n) ;
end proc:
seq(A360102(n),n=0..70) ; # R. J. Mathar, Mar 12 2023
  • PARI
    a(n) = sum(k=0, n, binomial(n+2*k, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)*(1+sqrt(1-4*x/(1-x)^3))))

Formula

G.f. A(x) satisfies A(x) = 1/(1-x) + x * A(x)^2 / (1-x)^2.
G.f.: (1/(1-x)) * c(x/(1-x)^3), where c(x) is the g.f. of A000108.
D-finite with recurrence (n+1)*a(n) +4*(-2*n+1)*a(n-1) +10*(n-2)*a(n-2) +2*(-2*n+7)*a(n-3) +(n-5)*a(n-4)=0. - R. J. Mathar, Mar 12 2023

A366695 G.f. satisfies A(x) = (1 + x)^3 + x*A(x)^2.

Original entry on oeis.org

1, 4, 11, 39, 166, 765, 3716, 18725, 96956, 512690, 2756806, 15027651, 82853678, 461215414, 2588619402, 14632777719, 83232244238, 476040155118, 2736005962314, 15793863291792, 91530881427964, 532343678619778, 3106141476531628, 18177446846299299
Offset: 0

Views

Author

Seiichi Manyama, Oct 16 2023

Keywords

Crossrefs

Cf. A025227, A366694.
Cf. A162481.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(3*(k+1), n-k)*binomial(2*k, k)/(k+1));

Formula

G.f.: A(x) = 2*(1+x)^3 / (1+sqrt(1-4*x*(1+x)^3)).
a(n) = Sum_{k=0..n} binomial(3*(k+1),n-k) * binomial(2*k,k)/(k+1).

A162482 Expansion of (1/(1-x)^3)*M(x/(1-x)^3), M(x) the g.f. of Motzkin numbers A001006.

Original entry on oeis.org

1, 4, 14, 53, 218, 945, 4235, 19441, 90947, 432030, 2078416, 10105435, 49578341, 245131321, 1220218293, 6110131376, 30756858405, 155547919269, 789965192900, 4027121386190, 20600180351659, 105707046807196, 543973305719611
Offset: 0

Views

Author

Paul Barry, Jul 04 2009

Keywords

Crossrefs

Cf. A001006, A162481.

Programs

  • Maple
    A162482 := proc(n)
    add(binomial(n+2*k+2,n-k)*A001006(k),k=0..n) ;
end proc:
seq(A162482(n),n=0..40) ; # R. J. Mathar, Feb 10 2015
  • Mathematica
    m[n_] := m[n] = If[n == 0, 1, m[n-1] + Sum[m[k]*m[n-2-k], {k, 0, n-2}]];
a[n_] := Sum[Binomial[n+2k+2, n-k]*m[k], {k, 0, n}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Apr 04 2024 *)

Formula

G.f.: 1/((1-x)^3-x-x^2/((1-x)^3-x-x^2/((1-x)^3-x-x^2/((1-x)^3-x-x^2/(1-... (continued fraction);
a(n) = Sum{k=0..n} C(n+2k+2,n-k)*A001006(k).
Conjecture: (n+2)*a(n) +4*(-2*n-1)*a(n-1) +18*(n-1)*a(n-2) +13*(-2*n+5)*a(n-3) +17*(n-4)*a(n-4) +3*(-2*n+11)*a(n-5) +(n-7)*a(n-6)=0. - R. J. Mathar, Feb 10 2015

A358518 a(n) = Sum_{k=0..n} binomial(n+3*k+3,n-k) * Catalan(k).

Original entry on oeis.org

1, 5, 20, 85, 405, 2116, 11766, 68237, 407789, 2492553, 15506942, 97859544, 624880895, 4029896310, 26209648212, 171711104853, 1132143259711, 7506530891217, 50019287312324, 334784759816729, 2249720564735567, 15172573979205166, 102662981205576494
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A086616, A162481, A360057.
Cf. A000108, A360046.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+3*k+3, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^4*(1+sqrt(1-4*x/(1-x)^4))))

Formula

a(n) = binomial(n+3,3) + Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^4 + x * A(x)^2.
G.f.: 2 / ( (1-x)^4 * (1 + sqrt( 1 - 4*x/(1-x)^4 )) ).
D-finite with recurrence (n+1)*a(n) +(-9*n+2)*a(n-1) +2*(7*n-4)*a(n-2) +10*(-n+2)*a(n-3) +5*(n-3)*a(n-4) +(-n+4)*a(n-5)=0. - R. J. Mathar, Jan 25 2023

A360058 a(n) = Sum_{k=0..n} (-1)^k * binomial(n+2*k+2,n-k) * Catalan(k).

Original entry on oeis.org

1, 2, 2, 2, 3, 3, 2, 4, 5, 0, 4, 13, -7, -7, 48, -16, -93, 180, 74, -584, 517, 1111, -2850, 207, 8281, -10738, -11740, 46967, -22167, -115845, 211052, 94468, -766989, 660110, 1554938, -3983408, 121429, 12272689, -15692006, -18841086, 72792247, -31828764
Offset: 0

Views

Author

Seiichi Manyama, Jan 23 2023

Keywords

Crossrefs

Cf. A360059, A360060.
Cf. A000108, A162481, A360049.

Programs

  • PARI
    a(n) = sum(k=0, n, (-1)^k*binomial(n+2*k+2, n-k)*binomial(2*k, k)/(k+1));
  • PARI
    my(N=50, x='x+O('x^N)); Vec(2/((1-x)^3*(1+sqrt(1+4*x/(1-x)^3))))

Formula

a(n) = binomial(n+2,2) - Sum_{k=0..n-1} a(k) * a(n-k-1).
G.f. A(x) satisfies A(x) = 1/(1-x)^3 - x * A(x)^2.
G.f.: 2 / ( (1-x)^3 * (1 + sqrt( 1 + 4*x/(1-x)^3 )) ).
D-finite with recurrence (n+1)*a(n) -2*a(n-1) +2*(n-3)*a(n-2) +4*(-n+2)*a(n-3) +(n-3)*a(n-4)=0. - R. J. Mathar, Jan 25 2023

A364624 G.f. satisfies A(x) = 1/(1-x)^3 + x*A(x)^4.

Original entry on oeis.org

1, 4, 22, 194, 2103, 25129, 318816, 4214724, 57419725, 800461033, 11363418314, 163708299724, 2387365301187, 35173224652637, 522752043513952, 7827979832083872, 117992516684761733, 1788819120580964014, 27258417705055812586, 417270970443908301926
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Crossrefs

Cf. A162481, A364623.

Programs

  • PARI
    a(n) = sum(k=0, n, binomial(n+8*k+2, 9*k+2)*binomial(4*k, k)/(3*k+1));

Formula

a(n) = Sum_{k=0..n} binomial(n+8*k+2,9*k+2) * binomial(4*k,k) / (3*k+1).

A364625 G.f. satisfies A(x) = 1/(1-x)^3 + x^2*A(x)^2.

Original entry on oeis.org

1, 3, 7, 16, 38, 95, 249, 678, 1901, 5451, 15906, 47066, 140868, 425657, 1296665, 3977684, 12276617, 38094013, 118768915, 371875752, 1168843808, 3686549845, 11664123048, 37011249678, 117750111763, 375529083267, 1200327617200, 3844662925222, 12338289374046
Offset: 0

Views

Author

Seiichi Manyama, Jul 30 2023

Keywords

Crossrefs

Cf. A000108, A086615, A162481, A360045.
Cf. A364626, A364627.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(2/((1-x)^3*(1+sqrt(1-4*x^2/(1-x)^3))))

Formula

G.f.: A(x) = 2 / ( (1-x)^3 * (1 + sqrt( 1 - 4*x^2/(1-x)^3 )) ).
a(n) = Sum_{k=0..floor(n/2)} binomial(n+k+2,3*k+2) * binomial(2*k,k) / (k+1).
Showing 1-10 of 12 results. Next

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